Discrete Applied Mathematics 156 (2008) 2480–2492

Contents lists available at ScienceDirect

Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

Identical part production in cyclic robotic cells: Concepts, overview andopen questionsNadia BraunerG-SCOP, UJF, 46 avenue Felix Viallet, 38031 Grenoble Cedex, France

a r t i c l e i n f o

Article history:Received 18 September 2006Received in revised form 27 June 2007Accepted 6 March 2008Available online 25 April 2008

Keywords:SchedulingFlow-shopMaterial handling systemCyclic production

a b s t r a c t

Robotic cells consist of a flow-shop with a robot for material handling. A single part is tobe produced cyclically and the objective is to minimize production rate. This documentintroduces basic concepts and tools for dealing with cyclic production. In particular, itconcentrates on k-cycles which are production cycles where exactly k parts enter and leavethe cell. One defines the cycle function K which is the smallest value of k so that the setof all k-cycles up to size K contains an optimal cycle for all instances. Known results andconjectures on these functions are given for the classical case where parts can remain onthemachinewaiting for the robot and for the no-wait casewhere parts have to be removedfrom the machine as soon as their processing is finished.

© 2008 Elsevier B.V. All rights reserved.

Robotic flow-shops consist ofmmachines served by a single central robot. They were first introduced by [4] and studiedby [40]. In [4], a line for machining castings for truck differential assemblies is described in the form of a 3-machine roboticcell where a robot has to transfer heavy mechanical parts between large machines. The system contains a conveyor belt forincoming parts and another one for outgoing parts. In this particular system the robot is not able to traverse the conveyor.Therefore, the movement of the robot from the output to the input station has to traverse the entire cell.

The original application has the form of a flow-shop. However, robotic cells may have very flexible configurations. Therobot can easily access the machines thus producing a large variety of products in form of a job-shop. But it is known thatthe robotic scheduling problem is already NP-hard for a flow-shop with m ≥ 3 machines and two or more different parttypes [28]. The case of the m-machine robotic cell in which one wants to produce a single batch of identical parts remainsof interest. We shall mainly concentrate on this case. The robot may have unit capacity, as will be the case in our model,or one may have two-unit robots [42,41] or a multi-robot cell [32,30]. Robotic cells with buffers at the machine have beenstudied in [24,13]. Operation and/or process flexibility (the order of the operations or the assignment of the operations tothe machines are not fixed) was recently studied in [3,27,26]. We concentrate here on the no-buffer case where parts areeither on a machine or transported by the robot and no flexibility on operations or on the process is allowed. A survey ongeneral robotic cells can be found in [18,23].

This document gives a state of the art on cyclic scheduling of identical parts in robotic cells. It describes classicalconfigurations of robotic cells (Section 1) and introduces basic concepts and tools for dealing with cyclic production(Section 2). Then it describes known results for the classical case where parts can remain on the machine waiting for therobot (Section 3) and for the no-wait case where the parts have to be removed from themachine as soon as their processingis finished (Section 4). Then a detailed list of open questions is given (Section 5).

1. Robotic cells

Themmachines of a robotic cell are denoted byM1,M2 . . .Mm andwe add two auxiliarymachines,M0 for the input stationIN andMm+1 for the output station OUT (Fig. 1). Rawmaterial for the parts to be produced is available in unlimited quantity

E-mail address: nadia.brauner@g-scop.inpg.fr.

0166-218X/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.dam.2008.03.021

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Elsevier - Publisher Connector

N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492 2481

Fig. 1. Robotic cell with m = 4 machines.

Table 1Two different expressions of the production constraints

Processing Waiting policy Another description

No-wait pi 0 pij = pijHSP in [pij, pij] 0 pij ≤ pijUnbounded pi Unbounded pij = +∞

at M0. The central robot can handle a single unit at a time. A part is picked up at M0 and transferred in succession to M1,M2 . . .Mm, where it is machined in this order until it finally reaches the output stationMm+1. AtMm+1, the finished parts canbe stored in unlimited amounts. We focus on the classical case as in [40], where machines M1, M2 . . .Mm are without bufferfacility. In this case, the robot, with unit capacity, has to be empty whenever it wants to pick up a part atMh (h = 0, 1 . . .m).

Consider an instance I of an m-machine robotic cell. Different cell configurations have been studied, depending mainlyon production constraints (Section 1.1) and on the metric for travel times (Section 1.2).

1.1. The production constraints

Processing starts as soon as a part is loaded on a machine. The processing time represents the minimum time a partmust remain on a machine. If all parts are different, pij denotes the processing time of part j on machine Mi. If one wants toproduce one large batch of identical parts, the processing times of the parts on machine Mi are pi = pij. In the balanced case,all processing times are equal, i.e., pi = p for all i.

Once the part is finished, two policies may apply. In the no-wait case, the part must be removed immediately from themachine and transferred to the following machine. In the unbounded case, the part can remain on the machine waiting forthe robot.

A classical extension of those two cases is the so-called Hoist Scheduling Problem (HSP) for which the processing policyis different. For the preceding two cases, the processing time is fixed. For the HSP, the processing time is described by aninterval, and the no-wait policy applies. Thismeans that the time part jmay remain onmachineMi lays in the interval [pij, pij]

(see Table 1). This applies to chemical treatments were themachines correspond to chemical baths. The HSP is NP-hard evenfor identical parts and very simple (additive) configurations of cells [20]. There exists a wide literature on this problem thatwe will not develop here (see e.g. [5]).

Denote by ε the time to load a part onto a machine from the robot or to unload a part from a machine onto the robot.

1.2. The travel metric of the robot

We shall consider different classical metrics for travel times of the robot depending on the physical configuration of thecell and on the characteristics of the robot. Denote by δh,h′ , the travel time of the robot (empty or loaded) fromMh toMh′ . Thefollowing natural, and in practice desirable, assumptions are made [14]:

• the travel time from a machine to itself is zero, that is, δh,h = 0;• the travel times satisfy the triangle inequality, that is, δh,k + δk,h′ ≥ δh,h′ for all h, k and h′;• The travel times are symmetric, that is δh,h′ = δh′,h for all h and h′.

Travel times verifying those three assumptions are called general (or sometimes “euclidean”, e.g. in [23]). Specialconfigurations of cells have been studied. Table 2 summarizes the most classical ones which we now define in detail.

For additive times, to travel between distant machines, the robot passes through all intermediate machines and its speedis constant. This metric is the most popular since, in practice, it is applicable if the machines are on a circle or on a line andif the cell is dense (the robot does not have time to speed up between distant machines). In this case, one has the triangleequality δh,h′ = δh,k + δk,h′ =

∑h′−1

k=h δk,k+1 for any h < h′. By symmetry, this also defines δh,h′ for h > h′. Some authors

2482 N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492

Table 2Some classical metrics for the robot travel times

δh,h+1 not constrained regular (δh,h+1 = δ)

General δh,h = 0; δh,h′ = δh′,h δh,h+1 = δ

δh,h′ ≤ δh,k + δk,h′

Additive δh,h′ = δh,k + δk,h′ for h < k < h′ δh,h′ = |h′− h|δ

Circular Shortest path along the circle δh,h′ = min(|h − h′|,m + 1 − |h − h′

|)

Constant δh,h′ = δ

(e.g. [29,35]) consider an extension of this case having a constant gain γ when travelling between distant machines, i.e.,δh,h′ =

∑h′−1

k=h δk,k+1 − (h′− h − 1)γ for h < h′. We assume γ = 0.

In circular cells, the input and the output stations coincide, i.e.M0 = Mm+1. Thus the robot chooses the shortest path alongthe circle formed by the machines. Then, the travel times verify δh,h′ = min(

∑h′−1

k=h δk,k+1,∑m

k=h′ δk,k+1 +∑h−1

k=0 δk,k+1) for anyh < h′.

In the regular case, machines are equidistant and we denote δh,h+1 = δ. This constraint can be added to additive (as inthe seminal paper [40]) or to circular cells.

For constant travel times (introduced in [22]), δ is the time for the robot to travel between any two distinct machinesMh and Mh′ : δh,h′ = δ. The interest of this metric is that it is simpler to study than the others but it seems to have the sameproperties as the general additive metric.

2. Activities and k-cycles

The robotic scheduling problem is already NP-hard for a flow-shop withm ≥ 3 machines and two or more different parttypes [28]. Therefore, we concentrate on the interesting case of the m-machine robotic cell in which one wants to produceidentical parts. Then the problem reduces to finding the optimal strategy for the robot moves in order to obtain themaximalthroughput rate for this unique part.

In [23], the authors prove that there always exists a cyclic production that is optimal. Therefore, we consider cyclic robotmoves for the production process of parts and define a k-cycle as a production cycle of exactly k parts. It can be described asa sequence of robot moves where exactly k parts enter the system atM0, k parts leave the system atMm+1 and each time therobot executes the k-cycle, the system returns to the same state, i.e. the same machines are loaded, the same machines areempty and the robot returns to the starting position. To describe k-cycles we use the concept of activities [21]. The activityAh (h = 0, 1 . . .m) consists of the following sequence:

– the idle robot takes a part from Mh;– the robot travels with this part from Mh to Mh+1;– the robot loads this part onto Mh+1.

Note, that many sequences are not feasible, e.g. (. . . A0, A0 . . .), since the robot carries a part to M1 which is occupied.In [21], the authors characterize k-cycles as follows: A k-cycle Ck is a sequence of activities, in which each activity occursexactly k times and between two consecutive (in a cyclic sense) occurrences of Ah(h = 1, 2 . . .m − 1) there is exactly oneoccurrence of Ah−1 and exactly one occurrence of Ah+1.

We represent a k-cycle Ck as in Fig. 2.The horizontal axis represents time. The vertical axis represents the cell. The graphindicates the position of the robot in the cell while executing the cycle. Dashed lines are empty robot moves and plainlines are loaded robot moves, the loading and unloading processes or the waiting times of the robot at the machines. Letus illustrate this with an example. In a 3-machine regular additive cell, consider the 1-cycle C = (A0A2A1A3). Let I be thefollowing instance:

δ = 1; ε = 0; p1 = 6; p2 = 9; p3 = 6.

At the beginning of the cycle C, machine M2 is loaded and machines M1 and M3 are empty. We suppose that, at time 0, thepart has been onM2 for 6 time units. At time 9, on Fig. 2, the robot is at machineM3 and is waiting one time unit for the partto be ready in order to execute activity A3. One can observe that the cycle does not repeat identically. In this example, if thepart was on M2 for 5.5 time units (instead of 6), the cycle would have repeated identically. However, the mean cycle time(14.5 in both case) does not seem to depend on the initial state (ergodicity). Let T(Ck) be the long run average execution timeof the k-cycle Ck.

We call T(Ck) the cycle time and T(Ck)/k the cycle length. The throughput rate is defined by k/T(Ck). Thus the ρ-cycle Cρ isoptimal if it maximizes throughput rate or equivalently minimizes cycle length T(Ck)/k over the set of all possible k-cycles(k = 1, 2, 3 . . .). A set of cycles S is said to be dominant if, for any instance, there exists a cycle of S that is optimal.

N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492 2483

Fig. 2. Representation of the 1-cycle C = (A0A2A1A3) for the instance I.

2.1. Dominant sets of cycles

Ideally, one would like to determine, for a given instance, an optimal k-cycle. However, this is so far not possible, exceptfor very particular cases, for instance for very slow or for very fast robots compared to processing times. In [40] the authorsproposed the following conjecture1-cycle Conjecture [40]: The set of 1-cycles is dominant. This conjecture is valid for 2-machine cells (see Section 2.4) andunbounded 3-machine cells [21,9]. However it is false for no-wait 3-machine cells [1] and unbounded regular additive orconstant 4-machine cells [11,23]. It has been replaced by the following conjecture:Agnetis’ Conjecture [1]: The set of k-cycles with k ≤ m − 1 is dominant.

Note that this conjecture was originally formulated for additive no-wait cells. Let SK be the set of all k-cycles with1 ≤ k ≤ K. We are interested in the minimal dominant set SK , i.e., SK is dominant and no SK ′ is dominant with K ′ < K . Wecan expect that K = K(m) is a function of the number of machines m. We call K(m) the cycle function.

Finding the optimal production cycle can be decomposed into three sub-problems that we address in the followingsections:

(P1) Determine K , i.e., find its constant value if it is indeed constant, lower bounds, finite upper bounds. . .(P2) Determine the complexity of finding the best cycle in Sk (where k can be any number between 1 and K).(P3) Determine the performance of Sk (where k can be any number between 1 and K) defined by

P (k) = maxinstances

best cycle length in Skbest cycle length in SK

.

Those three problems have been widely studied especially (P2) and (P3) for k = 1 when the 1-cycle Conjecture was stillopen. In the following sections, we review the state of the art for those problems for the unbounded case (Section 3) and forthe no-wait case (Section 4).

2.2. Bounds

In this section, we introduce several bounds on the cycle time. Consider a k-cycle Ck. We first introduce some inequalitiesthat only depend on the cycle and not on the configuration of the cell. Let mh(Ck) be the number of times the robot travelsbetween machines Mh and Mh+1 in both directions during one execution of the k-cycle Ck and let |S|Ck be the number ofoccurrences of the sequence of activities S in Ck and let ui(Ck) = |Ai−1Ai|Ck . For simplicity, we drop Ck in those notations whenno confusion is possible. If the robot never makes any dummy moves, one has [11,9]:

m0 = 2k (1)mm = 2k (2)m1 = 4k − 2u1 (3)mm−1 = 4k − 2um (4)m2 ≥ 4k − 2u2 − 2|A1A0A2| (5)mm−2 ≥ 4k − 2um−1 − 2|Am−2AmAm−1|. (6)

The following inequalities are often used as an optimality criterion. The intuition is that the lower bound is the time elapsedbetween two successive loadings of machine Mh (see Fig. 3). Since this happens k times in a cycle, we have a multiplicativefactor k.

T(Ck) ≥ k(δh,h+1 + δh+1,h−1 + δh−1,h + ph + 4ε) ∀h = 1, 2 . . .m. (7)

In the additive case it becomes,

T(Ck) ≥ k(2δh,h+1 + 2δh−1,h + ph + 4ε) ∀h = 1, 2 . . .m (8)

2484 N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492

Fig. 3. Illustration of inequalities (8) and (9).

Fig. 4. The state graph G3 .

and in the constant case one has,

T(Ck) ≥ k(3δ + ph + 4ε) ∀h = 1, 2 . . .m (9)

One of the interests of the constant case is the fact that travel time can be expressed easily: one has the following equalityfor travel time TT(Ck) of a k-cycle Ck:

TT(Ck) = 2k(m + 1)δ −

m∑i=1

uiδ. (10)

The intuition for this equality is that between two activities, one has a time δ if and only if the two activities are notconsecutive, i.e. they do not participate in a ui. This equality is proved for k = 1 in [22].

2.3. State graphs

At each instant in a production cycle, it is possible to calculate the part/machine incidence vector: machineMh is loaded ifand only if the next occurrence of Ah arrives before the next occurrence of Ah−1. For instance, at the beginning of the executionof the cycle (A0, A4, A6, A7, A5, A3, A2, A1), the part/machine incidence vector is (0, 1, 1, 1, 0, 1, 0).

In the state graph Gm associated with an m-machine robotic cell, each vertex is a part/machine incidence vector thatrepresents the state of the cell. Therefore, Gm has 2m vertices. Arcs represent the activities of the robot to pass from one stateto another state. The state graph G3 is given in Fig. 4. In G3, ‘100’ represents the state of the system with machine M1 loadedand machines M2 and M3 empty. To go from state ‘100’ to state ‘010’, the robot transfers a part from M1 to M2, executingactivity A1. Each k-cycle corresponds uniquely to a cycle of length k(m+1) in the graph. For instance, the 1-cycle (A0A3A1A2)corresponds to the sequence of vertices ‘001’, ‘101’, ‘100’, ‘010’.

State graphs allow us to find the number of k-cycles in an m-machine cell [15] (note that e.g. a 2-cycle might be therepetition of twice the same 1-cycle). The algorithm is based on the traversal of the state graph. A label is attached to eachcycle. This label is the inverse of the number of times the cycle appears during the traversal. One then obtains the number ofk-cycles by adding the labels of all the k-cycles encountered. Table 3 displays the number of k-cycles in an m-machine cell.One can remark that this number rapidly grows so that it is impossible to find the best production cycle by enumeration ofthe cycle times of all k-cycles.

N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492 2485

Table 3Number of k-cycles in an m-machine cell

k m2 3 4 5 6 7

1 2 6 24 120 720 50402 3 20 260 5588 175112 74390723 4 70 3656 375984 651172804 6 300 60648 292224245 8 1350 10736966 14 6580 198473167 20 326468 36 1666209 60 862470

Fig. 5. Definition of an arc in LGm .

Fig. 6. The line-graph LG3 of G3 .

To each graph Gm (with orientations) is associated a line-graph LGm with orientations as follows:

– the vertices of LGm correspond to the arcs of Gm;– (a, a′) is an arc of LGm if and only if there exists, in Gm, a vertex v which is the tail of a and the head of a′ as in Fig. 5.

Circuits of Gm and of LGm are equivalent. Therefore, one can work with either graph. The graph LG3 is represented onFig. 6. Arcs of LGm can be weighted by travel times of the robot or by some (unfortunately not all) waiting times. Therefore,a minimummean circuit length in LGm is a lower bound to the optimal cycle time.

Let G be an oriented graph with n vertices and a distance on the arcs. A minimum mean circuit C in G is a circuit whichminimizes

sum of the distances of the arcs of Cnumber of vertices in C

.

Finding aminimummean circuit is a problemwhich can be solved in a timepolynomial in the number of vertices of the graph(see e.g. [31,2]). Remark that the graph LGm has an exponential number of vertices ((m + 3)2m−2). However, this approachhas been used to prove many dominance results described in the following sections.

2.4. Special solved cases

In this section, we consider three very simple solved cases: a very slow or very fast robot (compared to processing times)and the 2-machine case.

Intuitively, when travel times aremuch larger than processing times, it can be interesting to carry out as few robotmovesas possible. In this case, it is optimal to enter a part in the cell and to let the robot transfer this part on all the machinessuccessively, waiting each time for the processing to complete. This is done by the 1-cycle π0 = (A0A1 . . . Am) also known asthe identity cycle. Its cycle time is

T(π0) =

m∑h=0

δh,h+1 + δm+1,0 + 2(m + 1)ε +

m∑h=1

ph.

2486 N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492

Table 4Cycle function, K(m), in the unbounded case (problem (P1) in Section 2.1)

Travel Production 2 3 4 5 ≤ m ≤ 15 m ≥ 16

General 1 ≥2Circular 1 ?a ?Additive or constant 1 ≥4aAdditive regular Balanced 1 ?Constant Balanced 1a Means “even for the regular case”.

Table 51-cycle complexity and performance in the unbounded case (problems (P2) and (P3) for k = 1 in Section 2.1)

Travel Production Complexity Performance

General NP-hard 4General Balanced NP-hard 4Circular ? ?Additive/constant Polynomial 1,5Additive Balanced O(1) ?Constant Balanced O(1) 1

On the other side, when the robot is very fast (compared to processing times), it is interesting to let it do manymoves whileother parts are being processed on the machines. In this case, the 1-cycle πd = (A0AmAm−1 . . . A1) also known as the downhillcycle is optimal. Its cycle time is

T(πd) = max(

m∑h=0

δh,h+1 +

m−1∑h=0

δh+2,h + 2(m + 1)ε + δ1,m;maxi

(pi + δi,i+1 + δi+1,i−1 + δi−1,i + 4ε))

.

Note that if T(πd) is equal to the second term of the max operator, then the lower bound in inequality (7) is attained.Therefore, in this case (large processing times), πd is optimal.

Another extreme problem is the 2-machine case. When the robot transfers a part from M1 to M2, both machines areempty. Moreover, between two consecutive occurrences of A1, one has A0A2 or A2A0. Therefore, a k-cycle can be decomposedinto k sequences of the two 1-cycles A1A0A2 and A1A2A0 and its cycle time is the sum of the cycle times of the sub-cyclescomposing it. Therefore, in this case, one-cycles are optimal (whatever the configuration of the cell is) and two 1-cycles arepossible: the identity and the downhill permutation. Just choose the best one.

3. The unbounded case

In the seminal paper [40], the authors consider additive regular robotic cells with unbounded waiting times at machinesand present the 1-cycle Conjecture. This section relates known results in the unbounded case, first describing the 9-year lifetime of the 1-cycle Conjecture for additive cells (Section 3.1). All results are then extended to the constant case(Section 3.2). Interest in 1-cycles and their simplicity induced research on their performance factor (Section 3.3). We thenconsider the well understood (but still partially open) regular balanced case for which the input is composed of only 4numbers (Section 3.4).

On our way, we describe the state of problems (P1), (P2) and (P3) (defined in Section 2.1) for different configurations ofunbounded robotic cells: Table 4 summarizes known values or bounds for the cycle functions K(m) depending on the cellconfigurations. Because of their simplicity, 1-cycles have generated a large amount of results. Table 5 shows the complexityof finding the best 1-cycles and the performance factor of 1-cycles.

3.1. The 1-cycle Conjecture for additive cells

In 1992, Sethi, Sriskandarajah, Sorger, Blazewicz and Kubiak [40] claim that the best production cycle can be achieved bya one cycle (i.e. K(m) = 1 for anym) in a regular additive cell (called the 1-cycle Conjecture). In the same paper, the authorsstate that this conjecture is true for 2-machine cells. In 1997, Hall, Kamoun and Sriskandarajah [29] prove, rather technically,that 1-cycles dominate 2-cycles in regular 3-machine cells. In 1999, Crama and van de Klundert [21] extend this result tok-cycles proving that the 1-cycle Conjecture is valid for 3-machine additive cells (a shorter proof based on the state graphG(3) and on inequalities (1) to (6) is given in [9]). Moreover, in regular additive 4-machine cells, 1-cycles again dominate2-cycles [6].

In 1997, interest in 1-cycles is enforced when Crama and van de Klundert [19] prove that the best 1-cycle can be foundin polynomial time. In [40], the authors prove that a 1-cycle is completely defined by a permutation of activities. Hence, thenumber of 1-cycles ism!. We consider, without loss of generality, that a 1-cycle π starts with activity A0. 1-cycles are then ofthe form π = (A0, Ai1 , Ai2 . . . Aim) where (i1, i2 . . . im) is a permutation of {1, 2 . . .m}. Let us consider 1-cycles π belonging tothe set of pyramidal permutations.We callπ = (A0, Ai1 , Ai2 . . . Aim)pyramidal if there is a p such that 1 ≤ i1 < · · · < ip = m and

N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492 2487

m > ip+1 > · · · > im ≥ 1. In [19], the authors prove that pyramidal permutations dominate 1-cycles. They give an algorithmof complexity O(m3) for the determination of the best pyramidal permutation which is therefore also the complexity offinding the best 1-cycle.

Unfortunately, the 1-cycle Conjecture happened to be false for additive regular 4-machine cells. Indeed, in 2001, Braunerand Finke [11] described a 3-cycle that is strictly better than all 1-cycles for a regular additive 4-machine cell thus provingthat K(4) ≥ 3 and reviving the question of the complexity of finding the best production cycle in robotic cells.

3.2. A metric unifying both the additive and the constant cases

All results presented in Section 3.1 are also valid for the constant case. Indeed, in [23], the authors claim that the graphicalproof of the validity of the 1-cycle Conjecture for 3-machine additive cells in [9], also works for the constant case. Weconjecture that the proof works for general 3-machine cells with travel times verifying δ12 + δ23 + δ40 ≥ δ24 + δ13 + δ02.

For the complexity of finding the best 1-cycle, define the basic cycles constructed as follow [22]:

– partition the activities A1, A2 . . . Am into two sets V1 and V2;– construct a sequence S of activities composed of A0 followed by the activities of V2 in decreasing order;– insert sequentially the activities of V1 in S in increasing order putting Ai ∈ V1 just after Ai−1 in S.

Let us illustrate this on an example: m = 8 and V1 = {A1, A2, A4, A8} and V2 = {A3, A5, A6, A7} make the 1-cycleA0A1A2A7A8A6A5A3A4. All 1-cycles that can be constructed as described above belong to the set of basic cycles (of cardinality2m

−m). In [22], the authors prove that in a constant robotic cell, basic cycles dominate 1-cycles and that the best basic cyclecan be found in polynomial time. Therefore, one has:

Theorem 1 ([19,22]). In additive or constant robotic cells, the best 1-cycle can be found in polynomial time.

In the quest for K(m), the 1-cycle Conjecture was replaced by Agnetis’ Conjecture that claims that K(m) ≤ m − 1. Thisconjecture was again proved to be false for 4-machine cells:

Proposition 2 ([12]). In a 4-machine cell, the 4-cycle

C4 = (A0A1A0A3A4A2A1A0A3A2A1A4A3A2A0A1A4A3A4A2)

strictly dominates all k-cycles for k = 1, 2, 3 for the following instance:

m = 4; δ = 1; ε = 0; p1 = 0; p4 = 0;

in the additive case : p2 = 10; p3 = 10;

in the constant case : p2 = 6; p3 = 6.

We conjecture that Proposition 2 is still valid with

– the followingmetric that generalizes the additive and the constant case: let δi be the time to travel between twomachineswith distances in the production process equal to i: δkl = δ|l−k|, for l, k = 0, 1 . . .m + 1 with the following assumptions

δi ≤ δj for i ≤ j and 0, 4 <δ2

δ1 + δ3< 1

– the same instance generalized to p2 = p3 = 3δ1 + 2δ2 + δ3.

Preceding extensions of the properties of the additive case to the constant case raise two natural questions:

• Is there a metric which generalizes both cases keeping known results on the optimality and the complexity of 1-cycles?• Is there a natural way to extend all proofs for the constant case (which seems simpler to study) to the additive case?

3.3. Performance of 1-cycles

Whenever the 1-cycle Conjecture is false, it is interesting to study the quality of 1-cycles since they are simple, easyto implement and the problem reduced to 1-cycles is polynomially solvable for most cases. Let us define the performancefactor of 1-cycles, P (1), as in Section 2.1:

P (1) = maxinstances

cycle length of the best 1-cyclecycle length of the best cycle in SK

.

2488 N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492

The 1-cycle πd allows to find the following bounds:

P (1) ≤

2 −δ0,1 + δm,m+1

δ0,1 + δm,m+1 +m−1∑i=1

δi,i+1

≤ 2 in the additive case [10,19];

P (1) ≤

(2 −

2m + 2

)≤ 2 in the constant case [23];

P (1) ≤ 4 in the general case [25].

For the additive regular case or for the constant case one has, P (1) ≤ 1.5 [25]. For a regular additive 4-machine robotic cell,one can give this value with more precision: We know that P (1) ≥ 16/15 with the example presented in Proposition 2.With a rather technical study, one may show that P (1) ≤ 9/8 and that this number is not tight.Finally, the reduced intervalfor P (1) in a regular additive 4-machine cell is given by [1.06666; 1.125].

3.4. Regular balanced cells

In this section, we consider balanced cells where an instance is given by four numbers:– the number of machines, m,– the travel time, δ (between consecutive machines in the additive case and between any two machines in the constant

case),– the loading/unloading time, ε,– the processing time, p.

This case is interesting since it raises complexity problems: the complete description of a production cycle (as a k-cyclefor instance) is not polynomial in the instance length (see [8] for a discussion on this topic). We shall discuss the complexityof finding the best 1-cycle and the optimality of 1-cycles for the constant (Section 3.4.1) and the additive (Section 3.4.2)cases.

3.4.1. ConstantFor the constant balanced case, the problem is easy. We prove that, if p < δ then the identity cycle π0 = (A0A1 . . . Am) is

optimal, otherwise, the downhill permutation πd = (A0AmAm−1 . . . A1) is optimal. If p = δ, then both cycles perform equallywell.

Theorem 3. In the constant balanced case, one has K(m) = 1 and the best 1-cycle can be found in constant time.

Proof. Consider a k-cycle Ck. Its cycle time is composed of travel times, loading/unloading times and waiting times. Eq. (10)indicates that the total travel time of Ck is 2k(m + 1)δ −

∑uiδ. Moreover, in Ck, each of the k × (m + 1) activities induces a

loading and unloading time of duration ε each. This leads to a total loading/unloading time of 2k(m + 1)ε. Moreover, whileexecuting a sequence Ai−1Ai, the robot transfers a part to Mi, waits at the machine during the process of the part and thentransfers this part to machine Mi+1. Therefore, each ui = |Ai−1Ai| generates a waiting time of p. Hence the cycle time of thek-cycle Ck satisfies

T(Ck) ≥ 2k(m + 1)δ +

m∑i=1

ui(p − δ) + 2k(m + 1)ε.

Moreover, the cycle times of the identity cycle and of the downhill permutation are

T(π0) = (m + 2)δ + mp + 2(m + 1)εT(πd) = 2(m + 1)δ + 2(m + 1)ε + max(0, p − (2m − 1)δ − 2(m − 1)ε).

If p ≥ (2m − 1)δ + 2(m − 1)ε, then πd achieves the lower bound given in inequality (9) and hence, πd is optimal. For thefollowing, we shall assume that p ≤ (2m − 1)δ + 2(m − 1)ε so that T(πd) = 2(m + 1)δ + 2(m + 1)ε.

If p ≥ δ then T(Ck) ≥ 2k(m + 1)δ + 2k(m + 1)ε = kT(πd) and πd is at least as good as Ck.If p ≤ δ then, since ui ≤ k and p − δ ≤ 0, one has

T(Ck) ≥ 2k(m + 1)δ + mk(p − δ) + 2k(m + 1)ε = k(m + 2)δ + 2k(m + 1)ε + mkp = kT(π0)

which proves that K(m) = 1. A polynomial time algorithm for the best 1-cycle would return π0 if p < δ and πd if p > δ andπ0 or πd for p = δ.

3.4.2. Additive regular caseIn additive regular balanced cells, the best 1-cycle can also be found in polynomial time [14]. We only consider the case

m ≥ 4, since for m ≤ 3, for the more general additive case, the 1-cycles are dominant and finding the best 1-cycle can bedone in polynomial time.

N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492 2489

Table 6Dominance and complexity results for the no-wait additive case

m Config. K(m) Complexity

2 Add. reg. 1 P [1]3 Add. reg. 2 P [1]4 Add. reg. ≥3 Open4 Reg. bal. 3 P [38]≥5 Reg. bal. ≥m − 1 Open (1-cycles in pol. time but not optimal)

Consider the m2 + 1 following 1-cycles for m ≥ 4 and m even:

π0 = (A0A1A2 . . . Am)

πα = (A0Aα+1Aα+3 . . . Am−α−1AmAm−1Am−2 . . . Am−αAm−α−2Am−α−4 . . . Aα+2AαAα−1Aα−2 . . . A2A1)

for α ∈

[1 . . .

m

2− 1

]πm/2 = (A0AmAm−1 . . . A2A1) = πd.

Theorem 4 ([14]). In an additive regular balanced cell, if m is even, the best 1-cycle isπ0 if 0 ≤ p ≤ δ;

πα if (4α − 3)δ + 2(α − 1)ε ≤ p ≤ (4α + 1)δ + 2αε; α ∈

[1 . . .

m

2− 1

];

π m2

if (2m − 3)δ + (m − 2)ε ≤ p.

The case with m odd is similar. Consider the following 1-cycles for m ≥ 5 and m odd :

π0 = (A0A1A2A3 . . . Am)

π10 = (A0A1A3 . . . Am−2AmAm−1Am−3 . . . A4A2)

π20 = (A0A2A4 . . . Am−1AmAm−2Am−4 . . . A3A1)

π1α = (A0Aα+1Aα+3 . . . Am−α−2AmAm−1Am−2 . . . Am−α−1Am−α−3Am−α−5 . . . Aα+2AαAα−1Aα−2 . . . A2A1)

for α ∈

[1 . . .

m − 32

]π2

α = (A0Aα+2Aα+4 . . . Am−α−1AmAm−1Am−2 . . . Am−αAm−α−2Am−α−4 . . . Aα+3Aα+1Aα−1Aα−2 . . . A2A1)

for α ∈

[1 . . .

m − 32

]π m−1

2= (A0AmAm−1 . . . A2A1) = πd.

The permutations π1α and π2

α have the same cycle time.

Theorem 5 ([14]). In an additive regular balanced cell, if m is odd, the best 1-cycle is

π0 if 0 ≤ p ≤ δ;

π10 if δ ≤ p ≤ 2δ;

π1α if (4α − 2)δ + 2(α − 1)ε ≤ p ≤ (4α + 2)δ + 2αε; α ∈

[1 . . .

m − 32

];

π m−12

if (2m − 4)δ + (m − 3)ε ≤ p.

Therefore, for a given instance, one just has to determineα fromm, p, δ, and ε to obtain the best 1-cycle. However, for additiveregular balanced cells, the exact value of K(m) is not completely known: for m ≤ 3, for the more general additive cells onehas K(m) = 1. For regular additive balanced cells, the detailed proof that 1-cycles are dominant can be found in [6] form = 4 and [7] for m ≤ 6. This last proof can be extended to m ≤ 15 with a case by case study. However, the value of K(m)with m ≥ 16 is still unknown.

4. The no-wait case

In this section, we review the no-wait case. Agnetis’ conjecture, which claims that K(m) = m − 1, was first formulatedfor this case. Table 6 summarizes known results on the value of K(m) and on the complexity of finding the best productioncycle for different values of m.

The no-wait case has historically been studied for additive regular cells. Form ≤ 3, Agnetis’ Conjecture was proved in [1].Form = 4, it has been proved for the balanced case (equal processing times on all machines) in [38]. The idea of the proof is

2490 N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492

Table 7Optimal cycles for regular balanced 4-machine cells

p Optimal cycle Degree Cycle length

[0, 4δ[ A0A1A2A3A4 1 4p + 10δ[4δ, 6δ[ A0A1A0A2A1A3A2A4A3A4 2 5p/2 + 7δ[6δ, 8δ[ A0A1A2A3A4 1 2p + 10δ[8δ, 10δ[ A0A3A2A1A4A3A2A4A3A0A4A1 2 5p/4 + 4δ[10δ, 12δ[ A0A4A3A1A0A4A2A1A0A3A2A1A4A3A2 3 4p/3+14δ/3≥ 12δ A0A4A3A2A1 1 p + 4δ

Table 8Optimal cycles for regular balanced 5-machine cells

p Minimal degree of an optimal cycle Optimal cycle length

[0, 4δ[ 1 5p + 12δ[4δ, 8δ[ 2 6p/2 + 8δ

[8δ, 10δ[ 3 7p/3 + 20δ/3

[10δ, 12δ[ 3 6p/3 + 18δ/3

[12δ, 14δ[ 4 6p/4 + 20δ/4[14δ, 16δ[ 4 5p/4 + 18δ/4≥ 16δ 1 p + 4δ

to use the state graph G4 and its line graph. The no-wait constraints allow, for each instance, to erase some forbidden arcs inthe graph. In the reduced graph, structural properties allow to prove the desired result. Table 7 describes the optimal cyclefor a regular balanced 4-machine cell. For simplicity, it considers the case ε = 0. The first column gives the correspondinginterval for the processing time p. Moreover, for m ≥ 4, the cycle function verifies K(m) ≥ m − 1. This was proved in [36]using the (m − 1)-cycle constructed below:

A0 Am Am−1 . . . A4 A3 A1A0 Am Am−1 . . . A4 A2 A1A0 Am Am−1 . . . A3 A2 A1

A0 Am Am−1... A4 A3 A2 A1

A0 Am . . . A4 A3 A2 A1A0 Am−1 . . . A4 A3 A2 A1

Am Am−1 . . . A4 A3 A2 .

For instance, for m = 5, this leads to the following 4-cycle:

A0A5A4A3A1A0A5A4A2A1A0A5A3A2A1A0A4A3A2A1A5A4A3A2.

Indeed, while executing this cycle, the robot transports m − 1 parts through the cell, but does not have the time (becauseof the no-wait constraint) to transport an additional part (which would have given the 1-cycle πd). Using this idea, in [39],a conjecture is given on a best (optimal and with minimum degree) k-cycle in a balanced cell. This conjecture, which goesa step further for solving Agnetis’ Conjecture, is still open but some special cases have been solved. Table 8 summarizes theopen cases (grey cells) in a regular balanced 5-machine cell (again with ε = 0).

In [39], it is also proved that, for each value of 1 ≤ k ≤ bm+14 c, there exists an optimal k-cycle that strictly dominates all

cycles with smaller degree. This implies that K(m) can not be bounded by a constant.In the additive no-wait case, finding the best 1-cycle or 2-cycle can be done in polynomial time [34,17]. However, the

preceding discussion indicates that 1- or 2-cycles are not optimal form ≥ 4. Therefore, the complexity of finding the optimalproduction cycle is still an open problem. If Agnetis’ Conjecture is true, this problem is reduced to finding the best k-cyclewith k ≤ m − 1.

Some extensions of this problem have been studied: In [37], the combined case with some machines with no-waitconstraints and others with unbounded waiting times is considered. For balanced 2- and 3-machine cells the completedominance table is known. For this case, K(2) = 1 and K(3) = 2 as for the no-wait case.

In [16,33], the authors present algorithms for finding the best k-cycle in the no-wait case for identical parts. Thosealgorithms are polynomial in m but exponential in k.

5. Open questions

This document has presented an overview of the scheduling problem of identical parts in a robotic cell. In this section, welist themain questions that are still open. For unboundedwaiting times, the 3-machine case is completely settled. Therefore,one can concentrate on the remaining questions on the 4-machine case (hoping that it will be possible to generalize theproofs to m machines):

N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492 2491

• Prove that K(4) = 4 in the additive regular or in the constant case: It is only known that, for certain instances, some4-cycles strictly dominate cycles with smaller degree, but the dominance of 4-cycles is not proved.

• Determine the exact performance factor of 1-cycles in a 4-machine cell.

More general questions also remain for additive or constant cells:

• Find the link between additive and constant cells (same results) or a metric generalizing them with the same properties(complexity of finding the best 1-cycle, identical K(m) for all sub-configurations. . . ).

• Determine lower and finite upper bounds for K .• Determine the complexity of finding the best k-cycle with k ≤ K(m) in an m-machine cell.

For special configurations, the two main questions are:

• Determine K(3) for circular cells: little work has been done on this configuration and even the 3-machine case is stillopen. This is linked to the preceding question that tries to generalize the metrics.

• Prove that K(m) = 1 for additive regular balanced cells and m ≥ 16; even for this very simple (4 number) problem, thedominance of 1-cycles is not proved.

The no-wait case seems to be easier to settle since the problem is muchmore constrained than the case with unboundedwaiting times. However, the problem of finding an optimal cycle is still open. The simpler configuration is the balancedproblem (with equal processing times on all machines). Therefore, we suggest to start with this case for the followingquestion and then to extend it to the general case.

• Prove (or refute) Agnetis’ conjecture. This reduces to proving that K(m) ≤ m − 1.

References

[1] A. Agnetis, Scheduling no-wait robotic cells with two and three machines, European Journal of Operational Research 132 (2) (2000) 303–314.[2] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows, Theory, Algorithms, and Applications, Prentice Hall, New Jersey, 1993.[3] M.S. Akturk, H. Gultekin, O.E. Karasan, Robotic cell scheduling with operational flexibility, Discrete Applied Mathematics 145 (3) (2005) 334–348.[4] C.R. Asfahl, Robots and Manufacturing Automation, John Wiley & Sons, New York, NY, 1985.[5] C. Bloch, Contribution à l’ordonnancement dynamique de lignes de traitement de surface, Thèse de doctorat, Université de Franche-Comté, U.F.R.

Sciences et Techniques, Besançon, 1999.[6] N. Brauner, Ordonnancement dans des cellules robotisées, Thèse de doctorat, Université Joseph Fourier, Grenoble, France, 1999.[7] N. Brauner, Y. Crama, G. Finke, One-unit production cycles in balanced robotic cells, in: Proceedings IEPM’01, International Conference on Industrial

Engineering and Production Management, Quebec City, Canada, 2001, pp. 508–518.[8] N. Brauner, Y. Crama, A. Grigoriev, J. van de Klundert, A framework for the complexity of high-multiplicity scheduling problems, Journal of

Combinatorial Optimization 9 (2005) 313–323.[9] N. Brauner, G. Finke, On a conjecture about robotic cells: New simplified proof for the three-machine case, Journal of Information Systems and

Operational Research - INFOR: Scheduling in Computer and Manufacturing Systems 37 (1) (1999) 20–36.[10] N. Brauner, G. Finke, Optimal moves of the material handling system in a robotic flow-shop, in: Proceedings IEPM’99, International Conference on

Industrial Engineering and Production Management, vol. 1, Glasgow, United Kingdom, 1999, pp. 409–417.[11] N. Brauner, G. Finke, Cycles and permutations in robotic cells, Mathematical and Computer Modelling 34 (5–6) (2001) 565–591.[12] N. Brauner, G. Finke, Robotic cells: Configurations, conjectures and cycle functions, in: Operations Research 2005, Bremen, Germany, 2005.[13] N. Brauner, G. Finke, C. Gueguen, Flow-shop robotisé monoproduit avec stockage, European Journal of Automation / Journal Européen des Systèmes

Automatisés - APII-JESA 32 (7–8) (1998) 875–891.[14] N. Brauner, G. Finke, W. Kubiak, Complexity of one-cycle robotic flow-shops, Journal of Scheduling 6 (4) (2003) 355–371.[15] N. Brauner, B. Vettier, Testing module for cyclic robotic cells, in: Proceedings MCPL’2000, Second Conference IFAC/IFIP/IEEE Management and Control

of Production and Logistics, vol. 2, Grenoble, France, 2000, pp. 723–728.[16] A. Che, C. Chu, F. Chu, Multicyclic hoist scheduling with constant processing times, IEEE Transactions on Robotic and Automation 18 (1) (2002) 69–80.[17] A. Che, C. Chu, E. Levner, A polynomial algorithm for 2-degree cyclic robot scheduling, European Journal of Operational Research 145 (1) (2003) 31–44.[18] Y. Crama, V. Kats, J. van de Klundert, E. Levner, Cyclic scheduling in robotic flowshops, Annals of Operations Research: Mathematics of Industrial

Systems 96 (2000) 97–124.[19] Y. Crama, J. van de Klundert, Cyclic scheduling of identical parts in a robotic cell, Operations Research 45 (6) (1997) 952–965.[20] Y. Crama, J. van de Klundert, Robotic flowshop scheduling is strongly NP-complete, in: O.J. Vrieze, W.K. Klein Haneveld, L.C.M. Kallenberg (Eds.), Ten

Years LNMB, CWI Tract 122, Amsterdam, The Netherlands, 1997, pp. 277–286.[21] Y. Crama, J. van de Klundert, Cyclic scheduling in 3-machine robotic flow shops, Journal of Scheduling 2 (1999) 35–54.[22] M. Dawande, C. Sriskandarajah, S. Sethi, On throughput maximization in constant travel-time robotic cells, Manufacturing and Service Operations

Management 4 (4) (2002) 296–312.[23] M. Dawande, H.N. Geismar, S.P. Sethi, C. Sriskandarajah, Sequencing and scheduling in robotic cells: Recent developments, Journal of Scheduling 8 (5)

(2005) 387–426.[24] G. Finke, C. Gueguen, N. Brauner, Robotic cells with buffer space, in: Proceedings Conference of the European Chapter on Combinatorial Optimization,

ECCO IX, Dublin, Ireland, 1996, page 9 pages unnumbered.[25] H.N. Geismar, M. Dawande, C. Sriskandarajah, Approximation algorithms for k-unit cyclic solutions in robotic cells, European Journal of Operational

Research 162 (2) (2005) 291–309.[26] H.N. Geismar, S.P Sethi, J.B. Sidney, C. Sriskandarajah, A note on productivity gains in flexible robotic cells, International Journal of Flexible

Manufacturing Systems 17 (1) (2005) 5–21.[27] H. Gultekin, M.S. Akturk, O.E. Karasan, Cyclic scheduling of a robotic cell with tooling constraints, European Journal Operational Research 174 (2)

(2006) 777–796.[28] H.G. Hall, H. Kamoun, C. Sriskandarajah, Scheduling in robotic cells: Complexity and steady state analysis, European Journal of Operational Research

109 (1998) 43–65.[29] N.G. Hall, H. Kamoun, C. Sriskandarajah, Scheduling in robotic cells: Classification, two and three machine cells, Operations Research 45 (3) (1997)

421–439.[30] I. Ioachim, E. Sanlaville, M. Lefebvre, The basic cyclic scheduling model for robotic flow shops, INFOR 39 (3) (2001) 257–271.[31] R.M. Karp, A characterization of the minimum cycle mean in a diagraph, Discrete Mathematics 23 (1978) 309–311.[32] V. Kats, E. Levner, Cycle scheduling in a robotic production line, Journal of Scheduling 5 (1) (2002) 23–41.

2492 N. Brauner / Discrete Applied Mathematics 156 (2008) 2480–2492

[33] V. Kats, E. Levner, L. Meyzin, Multiple-part cyclic hoist scheduling using a sieve method, IEEE Transactions on Robotic and Automation 15 (4) (1999)704–713.

[34] E. Levner, V. Kats, V.E. Levit, An improved algorithm for cyclic flowshop scheduling in a robotic cell, European Journal of Operational Research 97(1997) 500–508.

[35] R. Logendran, C. Sriskandarajah, Sequencing of robot activities and parts in two-machine robotic cells, International Journal of Production Research34 (12) (1996) 3447–3463.

[36] F. Mangione, Ordonnancement des ateliers de traitement de surface pour une production cyclique et mono-produit, Ph.D. Thesis, Institut NationalPolytechnique de Grenoble, 2003.

[37] F. Mangione, N. Brauner, B. Penz, Balanced hoist scheduling problem with unbounded or zero-width processing windows, European Journal ofAutomation / Journal Européen des Systèmes Automatisés - APII-JESA 37 (3) (2003) 391–404.

[38] F. Mangione, N. Brauner, B. Penz, Flow shop robotisé à quatre machines sans attente, in: Conférence Francophone de MOdélisation et SIMulation,MOSIM 03, Toulouse, France, 2003. pp. 542–545.

[39] F. Mangione, N. Brauner, B. Penz, Optimal cycles for the robotic balanced no-wait flow shop, in: Proceedings IEPM’03, International Conference ofIndustrial Engineering and Production Management, vol. 2, Porto, Portugal, 2003, pp. 539–547.

[40] S.P. Sethi, C. Sriskandarajah, G. Sorger, J. Blazewicz, W. Kubiak, Sequencing of parts and robot moves in a robotic cell, International Journal of FlexibleManufacturing Systems 4 (1992) 331–358.

[41] S.P. Sethi, J.B. Sidney, C. Sriskandarajah, Scheduling in dual gripper robotic cells for productivity gains, IEEE Transactions on Robotics and Automation17 (3) (2001) 324–341.

[42] Q. Su, F.F. Chen, Optimally sequencing of double-gripper gantry robot moves in tightly-coupled serial production systems, IEEE Transactions onRobotics and Automation 12 (1) (1996).